The proof of Theorem REF , similarly to the proof of the existence of modeling limits of plane trees in [1]},
has two steps: the decomposition step, focused on distilling first order properties of graphs in the sequence, and
the composition step, focused on constructing a modeling limit consistent with the identified first order properties.
These two steps also appear implicitly in [2]}, [3]},
in particular, the decomposition step is strongly related to the comb structure results presented in [2]}, [3]}.
The arguments of the decomposition step of the proof of Theorem REF  are analogous to those used in [1]}.
The composition step
however requires a conceptual extension of techniques used for modeling limits of trees as
we had to deal with vertex separations of sizes larger than one.
This was achieved by a careful analysis of different types of paths arising in the orientation corresponding to a weak coloring order.
This analysis allows defining the edge set of a modeling limit in a measurable and consistent way for vertex separations of sizes larger than one.
